3.1933 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^9} \, dx\)

Optimal. Leaf size=171 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{693 (d+e x)^7 \left (c d^2-a e^2\right )^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{99 (d+e x)^8 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^9 \left (c d^2-a e^2\right )} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11*(c*d^2 - a*e^2)*(d + e*x)^
9) + (8*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(99*(c*d^2 - a*e^2)^2
*(d + e*x)^8) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(693*
(c*d^2 - a*e^2)^3*(d + e*x)^7)

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Rubi [A]  time = 0.281855, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{693 (d+e x)^7 \left (c d^2-a e^2\right )^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{99 (d+e x)^8 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^9 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^9,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11*(c*d^2 - a*e^2)*(d + e*x)^
9) + (8*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(99*(c*d^2 - a*e^2)^2
*(d + e*x)^8) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(693*
(c*d^2 - a*e^2)^3*(d + e*x)^7)

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Rubi in Sympy [A]  time = 53.847, size = 160, normalized size = 0.94 \[ - \frac{16 c^{2} d^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{693 \left (d + e x\right )^{7} \left (a e^{2} - c d^{2}\right )^{3}} + \frac{8 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{99 \left (d + e x\right )^{8} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{11 \left (d + e x\right )^{9} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**9,x)

[Out]

-16*c**2*d**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(693*(d + e*x)**
7*(a*e**2 - c*d**2)**3) + 8*c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2
)/(99*(d + e*x)**8*(a*e**2 - c*d**2)**2) - 2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c
*d**2))**(7/2)/(11*(d + e*x)**9*(a*e**2 - c*d**2))

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Mathematica [A]  time = 0.204419, size = 104, normalized size = 0.61 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (63 a^2 e^4-14 a c d e^2 (11 d+2 e x)+c^2 d^2 \left (99 d^2+44 d e x+8 e^2 x^2\right )\right )}{693 (d+e x)^6 \left (c d^2-a e^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^9,x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(63*a^2*e^4 - 14*a*c*d*e^2*(11*
d + 2*e*x) + c^2*d^2*(99*d^2 + 44*d*e*x + 8*e^2*x^2)))/(693*(c*d^2 - a*e^2)^3*(d
+ e*x)^6)

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Maple [A]  time = 0.013, size = 146, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 8\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}-28\,xacd{e}^{3}+44\,x{c}^{2}{d}^{3}e+63\,{a}^{2}{e}^{4}-154\,ac{d}^{2}{e}^{2}+99\,{c}^{2}{d}^{4} \right ) }{693\, \left ( ex+d \right ) ^{8} \left ({a}^{3}{e}^{6}-3\,{a}^{2}c{d}^{2}{e}^{4}+3\,{c}^{2}{d}^{4}a{e}^{2}-{c}^{3}{d}^{6} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^9,x)

[Out]

-2/693*(c*d*x+a*e)*(8*c^2*d^2*e^2*x^2-28*a*c*d*e^3*x+44*c^2*d^3*e*x+63*a^2*e^4-1
54*a*c*d^2*e^2+99*c^2*d^4)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/(e*x+d)^8/(a^
3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/(e*x + d)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 13.3963, size = 768, normalized size = 4.49 \[ \frac{2 \,{\left (8 \, c^{5} d^{5} e^{2} x^{5} + 99 \, a^{3} c^{2} d^{4} e^{3} - 154 \, a^{4} c d^{2} e^{5} + 63 \, a^{5} e^{7} + 4 \,{\left (11 \, c^{5} d^{6} e - a c^{4} d^{4} e^{3}\right )} x^{4} +{\left (99 \, c^{5} d^{7} - 22 \, a c^{4} d^{5} e^{2} + 3 \, a^{2} c^{3} d^{3} e^{4}\right )} x^{3} +{\left (297 \, a c^{4} d^{6} e - 330 \, a^{2} c^{3} d^{4} e^{3} + 113 \, a^{3} c^{2} d^{2} e^{5}\right )} x^{2} +{\left (297 \, a^{2} c^{3} d^{5} e^{2} - 418 \, a^{3} c^{2} d^{3} e^{4} + 161 \, a^{4} c d e^{6}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{693 \,{\left (c^{3} d^{12} - 3 \, a c^{2} d^{10} e^{2} + 3 \, a^{2} c d^{8} e^{4} - a^{3} d^{6} e^{6} +{\left (c^{3} d^{6} e^{6} - 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} - a^{3} e^{12}\right )} x^{6} + 6 \,{\left (c^{3} d^{7} e^{5} - 3 \, a c^{2} d^{5} e^{7} + 3 \, a^{2} c d^{3} e^{9} - a^{3} d e^{11}\right )} x^{5} + 15 \,{\left (c^{3} d^{8} e^{4} - 3 \, a c^{2} d^{6} e^{6} + 3 \, a^{2} c d^{4} e^{8} - a^{3} d^{2} e^{10}\right )} x^{4} + 20 \,{\left (c^{3} d^{9} e^{3} - 3 \, a c^{2} d^{7} e^{5} + 3 \, a^{2} c d^{5} e^{7} - a^{3} d^{3} e^{9}\right )} x^{3} + 15 \,{\left (c^{3} d^{10} e^{2} - 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{6} e^{6} - a^{3} d^{4} e^{8}\right )} x^{2} + 6 \,{\left (c^{3} d^{11} e - 3 \, a c^{2} d^{9} e^{3} + 3 \, a^{2} c d^{7} e^{5} - a^{3} d^{5} e^{7}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/(e*x + d)^9,x, algorithm="fricas")

[Out]

2/693*(8*c^5*d^5*e^2*x^5 + 99*a^3*c^2*d^4*e^3 - 154*a^4*c*d^2*e^5 + 63*a^5*e^7 +
4*(11*c^5*d^6*e - a*c^4*d^4*e^3)*x^4 + (99*c^5*d^7 - 22*a*c^4*d^5*e^2 + 3*a^2*c
^3*d^3*e^4)*x^3 + (297*a*c^4*d^6*e - 330*a^2*c^3*d^4*e^3 + 113*a^3*c^2*d^2*e^5)*
x^2 + (297*a^2*c^3*d^5*e^2 - 418*a^3*c^2*d^3*e^4 + 161*a^4*c*d*e^6)*x)*sqrt(c*d*
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^3*d^12 - 3*a*c^2*d^10*e^2 + 3*a^2*c*d^8*e^
4 - a^3*d^6*e^6 + (c^3*d^6*e^6 - 3*a*c^2*d^4*e^8 + 3*a^2*c*d^2*e^10 - a^3*e^12)*
x^6 + 6*(c^3*d^7*e^5 - 3*a*c^2*d^5*e^7 + 3*a^2*c*d^3*e^9 - a^3*d*e^11)*x^5 + 15*
(c^3*d^8*e^4 - 3*a*c^2*d^6*e^6 + 3*a^2*c*d^4*e^8 - a^3*d^2*e^10)*x^4 + 20*(c^3*d
^9*e^3 - 3*a*c^2*d^7*e^5 + 3*a^2*c*d^5*e^7 - a^3*d^3*e^9)*x^3 + 15*(c^3*d^10*e^2
- 3*a*c^2*d^8*e^4 + 3*a^2*c*d^6*e^6 - a^3*d^4*e^8)*x^2 + 6*(c^3*d^11*e - 3*a*c^
2*d^9*e^3 + 3*a^2*c*d^7*e^5 - a^3*d^5*e^7)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**9,x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/(e*x + d)^9,x, algorithm="giac")

[Out]

Timed out